Mean Deviation About Mean – The Intuition First
Imagine you have a small set of numbers: the marks of five students in a test: 4, 6, 8, 10, 12. The average (mean) is 8. Now, each student is some distance away from this average. The student who scored 4 is 4 marks below the mean; the one who scored 12 is 4 marks above. The student who scored 8 is exactly at the mean.
If you simply add these distances, the positives and negatives cancel out — you get zero. That's not useful. So instead, we ask: on average, how far is each data point from the mean? That's the mean deviation about mean.
Mean deviation is a measure of spread or dispersion. It tells you how scattered the data is around the central value. A small mean deviation means most data points are close to the mean; a large one means they are spread out.
The Precise Definition
For a set of n observations x1,x2,…,xn with mean xˉ, the mean deviation about mean (often written as MD or M.D.) is:
MD(xˉ)=n1∑i=1n∣xi−xˉ∣
That vertical bars mean absolute value — we take the distance without caring about direction. So every deviation is positive.
Mean Deviation about Mean=n∑∣xi−xˉ∣
Step-by-Step Calculation
Let's use the marks example: 4, 6, 8, 10, 12.
Step 1: Find the mean.
xˉ=54+6+8+10+12=540=8
Step 2: Find each absolute deviation ∣xi−xˉ∣.
| xi | xi−xˉ | ∣xi−xˉ∣ |
|------|----------------|-------------------|
| 4 | -4 | 4 |
| 6 | -2 | 2 |
| 8 | 0 | 0 |
| 10 | 2 | 2 |
| 12 | 4 | 4 |
Step 3: Sum the absolute deviations.
4+2+0+2+4=12
Step 4: Divide by n=5.
MD=512=2.4
So, on average, each student's mark is 2.4 marks away from the mean of 8.
Notice that the mean deviation is always less than or equal to the standard deviation (another measure of spread). For this data, standard deviation is about 2.83, which is larger than 2.4. This is because standard deviation squares deviations, giving more weight to extreme values.
Why Use Absolute Values?
You might wonder: why not just average the plain deviations (without absolute value)? Because the sum of (xi−xˉ) is always zero — that's a property of the mean. The absolute value is the simplest way to make all deviations positive so they don't cancel.
A common mistake: forgetting to take absolute values and getting zero. Always check: if your sum of deviations is zero, you forgot the absolute value.
When Is This Used?
Mean deviation is intuitive and easy to explain. It's used in:
- Quality control (checking how consistent a manufacturing process is) …